Use principles of force and kinematics to answer these:

a.
A 40 kg skater pushes on the 60 kg skater (initially at rest). As they collide, the 40 kg skater pushes on the 60 kg skater, applying a force 100 Newtons for 1.2 seconds. Find the velocity at which each of them ends up moving (they should be equal).
b.
A 60kg skater slides at 3.0 m/s toward a 40 kg skater (initially at rest). They collide and hold on, so both have the same final speed. find that speed. You don't need to know the Force and the time, it will work out.

a. To find the velocities of the skaters after the collision, we can use the principles of force and kinematics.

First, let's find the acceleration of the skaters using Newton's second law of motion:

Acceleration (a) = Force (F) / Mass (m)

For the 40 kg skater:
Acceleration (a1) = Force / Mass
= 100 N / 40 kg
= 2.5 m/s^2

For the 60 kg skater:
Acceleration (a2) = Force / Mass
= 100 N / 60 kg
= 1.67 m/s^2

Next, we can use the kinematic equation to find the final velocities of the skaters:

Final velocity (v) = Initial velocity (u) + (acceleration * time)

Since the initial velocity of both skaters is zero, the equation simplifies to:
Final velocity (v) = acceleration * time

For the 40 kg skater:
Final velocity (v1) = a1 * time
= 2.5 m/s^2 * 1.2 s
= 3 m/s

For the 60 kg skater:
Final velocity (v2) = a2 * time
= 1.67 m/s^2 * 1.2 s
= 2 m/s

Therefore, the velocity at which each skater ends up moving is 3 m/s.

b. In this case, since both skaters hold on to each other after the collision and have the same final speed, we can apply the principle of conservation of momentum.

The law of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision, assuming no external forces are present.

The momentum (p) of an object is given by the product of its mass (m) and velocity (v):

Momentum (p) = Mass (m) * Velocity (v)

Before the collision, the momentum of the 60 kg skater is:
Initial momentum (p1) = Mass * Initial velocity
= 60 kg * 3.0 m/s
= 180 kg·m/s

Since the 40 kg skater is initially at rest, their initial momentum (p2) is zero.

After the collision, both skaters have the same final speed. Let's assume this speed is v final.

The momentum of the 60 kg skater after the collision is:
Final momentum (p1') = Mass * Final velocity1
= 60 kg * v final

The momentum of the 40 kg skater after the collision is:
Final momentum (p2') = Mass * Final velocity2
= 40 kg * v final

According to the conservation of momentum:
Initial momentum (p1) + Initial momentum (p2) = Final momentum (p1') + Final momentum (p2')

180 kg·m/s + 0 kg·m/s = 60 kg * v final + 40 kg * v final

Simplifying the equation:
180 kg·m/s = 100 kg * v final

Dividing both sides by 100 kg:
v final = 1.8 m/s

Therefore, the final speed at which both skaters move after the collision is 1.8 m/s.

a. To find the velocities at which each skater ends up moving, we can use the principles of force and kinematics.

First, let's calculate the acceleration of the system using Newton's second law of motion, which states that the force applied is equal to the mass multiplied by the acceleration. The force applied by the 40 kg skater on the 60 kg skater is 100 Newtons. So,

Force = mass × acceleration

100 N = 60 kg × acceleration

Solving for acceleration, we get:

acceleration = 100 N / 60 kg

acceleration ≈ 1.67 m/s²

Now, using the kinematic equation:

final velocity = initial velocity + (acceleration × time)

As both skaters were initially at rest, their initial velocities are zero. Plugging in the values, we get:

final velocity = 0 + (1.67 m/s² × 1.2 s)

final velocity ≈ 2 m/s

Therefore, both skaters will end up moving with a velocity of approximately 2 m/s.

b. In this scenario, we don't need to know the force or the time involved, as both skaters have the same final speed. We can determine the final speed by applying the principle of conservation of momentum.

The momentum before the collision is given by:

momentum before = mass1 × velocity1 + mass2 × velocity2

Since one skater is initially at rest (velocity2 = 0) and the other skater is moving at a velocity of 3.0 m/s (velocity1 = 3.0 m/s), the equation simplifies to:

momentum before = 60 kg × 3.0 m/s + 0 kg × 0 m/s

momentum before = 180 kg·m/s

According to the principle of conservation of momentum, the total momentum before the collision should be equal to the total momentum after the collision, as long as no external forces are acting on the system.

momentum before = momentum after

mass1 × final velocity + mass2 × final velocity = momentum before

(60 kg + 40 kg) × final velocity = 180 kg·m/s

100 kg × final velocity = 180 kg·m/s

Now we can solve for the final velocity:

final velocity = 180 kg·m/s / 100 kg

final velocity = 1.8 m/s

Therefore, the final speed of both skaters will be approximately 1.8 m/s after the collision.